Category:Definitions/Variations in Sign of Polynomial

This category contains definitions related to Variations in Sign of Polynomial.
Related results can be found in Category:Variations in Sign of Polynomial.

Let $\map P x$ be a polynomial equation over the real numbers:

$\pm a_n x^n \mathop \pm a_{n - 1} x^{n - 1} \mathop \pm \cdots \mathop \pm a_1 x \mathop \pm a_0 = 0$

where:

the independent variable $x \in \R$ is real

the coefficients $a_j \in \R_{\ge 0}$ are all non-negative constants

each of the $\pm$ signs can be either positive or negative.

A variation in sign of the coefficients of $\map P x$ is an instance of $j \in \set {0, 1, \ldots, n}$ in which a $\pm$ sign changes from positive to negative, or vice versa, as $j$ goes from $n$ to $0$, ignoring zero coefficients.

That is, it is a change of sign in the tuple $\tuple {a_n, a_{n - 1}, \ldots, a_0}$.